If you didn't grab a copy of the partial fractions handout last week in class, you can find it here. It goes through the steps of partial fractions decompositions, and discusses how you can use the cover-up method for finding the coefficients (A, B, ...) quickly.

This week on Wednesday or Thursday I'm planning to hold a review session for Monday's midterm. If you want to give input on what times work well for you, here's a survey where you can.

Today we finished up the example from last Friday, and then started on another example with a system driven by a periodic external force F(t) = cos ωt. We set up the equation for the system and solved it.

Here are links to two applets written by Gautam Sisodia, another 307 instructor. The first allows you to see the solution for unforced systems and vary the m, γ, and k values and initial conditions: link.

The second graphs the solution for driven systems with a periodic external force F(t) = G sin ωt: link. The seven sliders, in order, control m, γ, k, y(0), y'(0), G, and ω.

Today we began talking about the applications of second-order linear equations, and in particular the spring-mass setup, which was a simple model for different kinds of real-life oscillations. In this setup, a mass hangs from a spring, possibly experiencing some damping force due to air resistance, and possibly experiencing additional external forces from outside the system.

During this class, we started analyzing the motion in this setup when there is no external force (unforced) and no damping (undamped). We started on an example with a verbal description of the system, first setting up an initial value problem then solving it. Finally, we started talking about how to convert the solution of the ODE into a more useful form (standard sinusoidal form). The standard sinusoidal form makes it easier to answer questions about the motion of the system, like how much the mass moves up and down, how quickly it oscillates, and when it first passes the equilibrium point. This handout describes standard sinusoidal form and the conversion process briefly.

The book covers these topics in section 3.7.

In this class we talked about one problem that can come up with the method of undetermined coefficients the way we've done it up until now. As an example, we tried to solve

y'' - 3y' + 2y = e^t.

Based on the right-hand side, the trial solution should be Y(t) = Ae^t, but if we plug this into the left-hand side, we get zero, not e^t. The problem is that Ae^t is a solution to the homogeneous equation y'' - 3y' + 2y = 0. The solution is to modify the trial solution by multiplying Ae^t by t first.

The general rule is: if X is a part of the trial solution that's also a solution to the homogeneous, then multiply X by t. If the trial solution also includes terms that are powers of t times X, they'll need to be multiplied by t also. For instance, if the homogeneous equation has e^t as a solution, and Y(t) = Ae^t + Bte^t + Ct²e^t + D , the A, B and C terms would each have to be multiplied by t. On the other hand, the D term we wouldn't touch, because it isn't a solution to the homogeneous equation.

Today's class we recapped the process we use to find the trial solution Y(t) from the nonhomogeneous part of the equation. For the rest of class, we talked about how to find the general solution of a nonhomogeneous equation by adding one particular solution of the nonhomogeneous equation to the general solution of the corresponding homogeneous equation. For instance, if we're given the equation y'' + 3y' + 2y = 4 to solve, the method of undetermined coefficients says that one solution is Y(t) = 2. The general solution to the corresponding homogeneous equation, y'' + 3y' + 2y = 0, is y_c(t) = ce^-t + de^-2t. Adding these two together will give us the general solution to the nonhomogeneous equation:

y(t) = Y(t) + y_c(t) = ce^-t + de^-2t + 2.

Today we had a quiz on SLHCCs, and then began talking about nonhomogeneous constant-coefficient equations, which look like

ay'' + by' + cy = g(t)

where a, b, and c are constants, and g(t) is a function.

These often come up in motion problems when there is an external force applied to a system (or in circuit problems, where there is an external voltage source). We began talking about one way to solve these, the method of undetermined coefficients, which is an educated guess-and-check method. The idea with undetermined coefficients is to take the nonhomogeneous part of the equation (this is g(t), the part that doesn't involve y or its derivatives) and use it to guess what one solution to the equation looks like.

Our textbook covers undetermined coefficients in section 3.5. If you read it, you'll note the book has a different approach to finding the trial solution, where you look up the appropriate trial solution in a table. You're welcome to learn and use this approach in your work and on exams, but make sure to make a note of it if you do.

Today we talked about how to solve SLHCCs when the characteristic equation had roots that were complex numbers (they involve the imaginary number i). Complex numbers enter the scene when the discriminant, the number under the square root in the quadratic formula, is negative. To find the general solution in this case, we could still use the technique from Friday using y = e^rt, but we have to figure out what e^rt means when r is a complex number. Here Euler's formula comes in:

e^iu = cos(u) + i sin(u)

Using Euler's method, and by taking special linear combinations of the solutions, we found that if the roots are r = a + bi, r = a - bi, then the general solution can be written as y = ce^atcos(bt) + de^atsin(bt), where c, d are arbitrary constants. In the book, this is covered in section 3.3.

We'll be using complex numbers in class today, and making sense of e^x when x is a complex number. Here are two handouts on complex numbers if you'd like a refresher on how complex numbers work: first handout, second handout.

Today we talked about how to solve 2nd-order linear homogeneous constant coefficient ODEs (SLHCCs for short) when the characteristic equation has a repeated root. Using the method from the last class (plugging in y = e^rt and solving for r) we only get one solution instead of two because of the repeated root (section 3.4 in the book).

To find a second solution we tried a new technique called reduction of order, which is another substitution method. If y₁(t) is some function we already know is a solution to the ODE (e^rt in this case) then we substitute a new variable v(t) for y, which we do by plugging in y = v(t)y₁(t). This gives us a new second-order ODE for v(t) that is easier to solve. Once we solve for v(t), we can get a new solution y(t) for the original ODE by multiplying v(t) by y₁(t)

Today we're starting into the second major topic of Math 307: linear second-order ODEs (with constant coefficients), which is chapter 3 in the textbook.

In this class we'll begin talking about homogeneous linear second-order ODEs, how to solve them, and hopefully get to initial conditions for second-order ODEs (In the textbook, this is covered in section 3.1).

Today in class we did some review for the midterm.

We'll be having review for the midterm today in office hours (4:30 – 5:30, in the Math Study Center) and tomorrow, 4 – 6, location TBD. I'll send out the location for tomorrow's review session by email, so stay tuned!

The first midterm will be taking place on Wednesday, April 24th in class. I will send out practice midterms soon, and next week we will have review in class Monday, as well as a review session outside of class.

If you're interested in coming to the review sesion, please take a moment to fill out this survey on what times work best for you. Thanks!

For many ODEs, we can't write down a formula for the solution. Today we talked about Euler's method, which is an algorithm to generate approximate solutions to an ODE when we can't find the solution exactly (This is covered in section 2.7 in the book).

Here are links to the two applets we used in class today: applet 1, applet 2. Here's another useful applet on Euler's method: link.

Today, we found a quick formula for finding the integrating factor for a linear first-order ODE. Before, we had to solve another ODE (which was separable) to find the integrating factor. The book talks about this in section 2.1, on page 36.

Then, we talked about solving Bernoulli equations, which were nonlinear 1st-order equations which look like p(t)y' + q(t)y = g(t)yⁿ. To solve these, we used a substitution, v(t) = y^1-n. This substitution transforms the Bernoulli equation into a linear ODE, which we now have to solve for v(t). Once we know v(t), we can undo the substitution to find y(t). The book discusses Bernoulli equations in homework problems 27–31 for section 2.4.

In this class, Christian Rudnick did another example of analyzing an autonomous equation like you did on Friday, and then solved the logistic equation using separation of variables to check that our new techniques really do work. (textbook: section 2.5 again).

During this class, Tvrtko Tadić introduced autonomous equations, differential equations where dy/dt depends only on y and not on t. He introduced how to analyze these equations and their solutions without actually solving the equations (textbook: section 2.5).

The plan for this class is to continuing talking about applications of ODEs, including another mixing problem and potentially a motion problem. (In the textbook, this is covered in section 2.3.)

We did a warm-up, reviewing the two techniques for solving first-order ODEs from the first week of class (separation of variables and integrating factors). Afterwards we began talking about applications of 1st-order ODEs, starting with mixing problems (section 2.3 in the textbook).

On Friday the plan is to have a short quiz on integrals (focusing on u-substitution and integration by parts, like the review problems). After the quiz, we can go on to solving 1st-order linear equations and a new technique (integrating factors) which we can use to solve them.

Here are some extra examples of solving linear equations.

Here's a video made by Nathan Grigg, another 307 instructor, which shows the process of solving 1st-order linear equations, so you can get a head start on Friday's class. Thanks Nathan!

Here are answers for the review integrals sheet.

We talked about different ways of classifying DEs: by order, ordinary vs. partial, single equations vs. systems, linear vs. nonlinear, and homogeneous vs. nonhomogeneous. The book discusses this in section 1.3, pages 19-21 (starting from "Classification of Differential Equations" and ending at figure 1.3.1). After that, we started talking about separable equations (section 2.2 in the book) and did an example of solving them.

I'll try to announce the topics for each class ahead of time like this on the website, and include the relevant sections from the book if you need them.

We went over the syllabus and talked about what differential equations are, why we study them, and did an example involving a falling object with drag. We also talked about using direction fields to visualize solutions to first-order ODEs. Some of the material you can find in the book, in section 1.1.

Here's a link to the syllabus and several review integrals. We'll have a short (10-minute) quiz on review integrals Friday.

Also, please take a moment to fill out this survey to help me choose our regular office hours for the quarter. Thank you!